Generalized Hilbert matrices and hypergeometric functions

Generalized Hilbert matrices and hypergeometric functions

joint with A. Montes Rodriguez and A. Sarafoleanu

May 30, 2010

A generalized Hilbert matrix has the form H_λ=

1 m+n+λ

m,n∈N∪{0}

whereλ∈C\ {0,−1,−2. . . ,}.

This is a classical Hankel matrix which has attracted a lot of attention.

The famous Hilbert inequality

kH₁k_l2→l² ≤π;

Recent generalization to weightedl²-spaces by Diamatopoulos - Siskakis;

”Trick or treats with the Hilbert matrix” by Man-Deun Choi.

This is a classical Hankel matrix which has attracted a lot of attention.

The famous Hilbert inequality

kH₁k_l2→l² ≤π;

Recent generalization to weightedl²-spaces by Diamatopoulos - Siskakis;

”Trick or treats with the Hilbert matrix” by Man-Deun Choi.

This is a classical Hankel matrix which has attracted a lot of attention.

The famous Hilbert inequality

kH₁k_l2→l² ≤π;

Recent generalization to weightedl²-spaces by Diamatopoulos - Siskakis;

”Trick or treats with the Hilbert matrix” by Man-Deun Choi.

Ifλ∈R,H_λ induces a bounded linear operator onl²that is self adjoint.

Its spectrum, and especially the eigenvalue problem H_λx =µx

have been intensively studied in the 50’s.

More general version of the eigenvalue problem:

µ∈Cis called alatent rootif there exists a non zero sequence {x_n}that satisfies

N

X

n=0

xn

n+m+λ −→µxm.

Its spectrum, and especially the eigenvalue problem H_λx =µx

have been intensively studied in the 50’s.

More general version of the eigenvalue problem:

µ∈Cis called alatent rootif there exists a non zero sequence {x_n}that satisfies

N

X

n=0

xn

n+m+λ −→µxm.

In 1957 T. Kato shows thatπis a latent root forH₁(affirmative answer to a question raised by O. Taussky in ’49).

The corresponding eigenvector is a positive sequence, but it is not inl²!

In fact he shows the same result holds forH_λwhenλ∈Rand λ≥ ¹₂.

In 1958 M. Rosenblum proves that every complex numberµ with Reµ >0 is a latent root ofH_λ, λ∈Rwithλ >0.

The result actually holds for allλ∈C,λ6=0,−1,−2, . . . ,and it was proved by C.K. Hill in 1960.

In 1957 T. Kato shows thatπis a latent root forH₁(affirmative answer to a question raised by O. Taussky in ’49).

The corresponding eigenvector is a positive sequence, but it is not inl²!

In fact he shows the same result holds forH_λwhenλ∈Rand λ≥ ¹₂.

In 1958 M. Rosenblum proves that every complex numberµ with Reµ >0 is a latent root ofH_λ, λ∈Rwithλ >0.

The result actually holds for allλ∈C,λ6=0,−1,−2, . . . ,and it was proved by C.K. Hill in 1960.

The spectrum onl².

Ifλ∈[¹₂,∞)then the point spectrum ofH_λ onl²is void and its spectrum is[0, π](W. Magnus, 1950 forλ=1 and M.

Rosenblum 1958)

Ifλ∈(−∞,¹₂)andpandqare the largest nonnegative integers such that 2p< ¹₂−λand 2q <−¹₂ −λthenπ/sinπλand

−π/sinπλare eigenvalues ofH_λ|l²of multiplicitiesp+1 and q+1. H_λhas no other eigenvalues. The spectrum is

[0, π]∪ {pointspectrum}(Rosenblum 1958).

The spectrum onl².

Ifλ∈[¹₂,∞)then the point spectrum ofH_λ onl²is void and its spectrum is[0, π](W. Magnus, 1950 forλ=1 and M.

Rosenblum 1958)

Ifλ∈(−∞,¹₂)andpandqare the largest nonnegative integers such that 2p< ¹₂−λand 2q <−¹₂ −λthenπ/sinπλand

−π/sinπλare eigenvalues ofH_λ|l²of multiplicitiesp+1 and q+1. H_λhas no other eigenvalues. The spectrum is

[0, π]∪ {pointspectrum}(Rosenblum 1958).

On Rosenblum’s approach

We move froml²toL²((0,∞))by mapping the standard orthonormal basis inl²onto the basis formed by

φn(t) =e^−t/2Ln(t), whereLn denotes the normalizedn-th Laguerre polynomial.

Under this transformationH_λ becomes an integral operator with kernel

K(x,t) = Γ(λ)W_1−λ,1/2(x+t)(x +t)⁻¹, whereW_1−λ,1/2denotes the Whittaker function.

Using some special identities satisfied by these functions Rosenblum proves that this operator has eigenfunctions in L¹((0,∞))of the formx →W_1−λ,µ(x)x⁻¹, which gives latent eigenvectors inl²of the form(x_n)with

x = Z ∞

W (x)φ (x)dx

. (1)

On Rosenblum’s approach

We move froml²toL²((0,∞))by mapping the standard orthonormal basis inl²onto the basis formed by

φn(t) =e^−t/2Ln(t), whereLn denotes the normalizedn-th Laguerre polynomial.

Under this transformationH_λ becomes an integral operator with kernel

K(x,t) = Γ(λ)W_1−λ,1/2(x+t)(x +t)⁻¹, whereW_1−λ,1/2denotes the Whittaker function.

Using some special identities satisfied by these functions Rosenblum proves that this operator has eigenfunctions in L¹((0,∞))of the formx →W_1−λ,µ(x)x⁻¹, which gives latent eigenvectors inl²of the form(x_n)with

x = Z ∞

W (x)φ (x)dx

. (1)

On Rosenblum’s approach

We move froml²toL²((0,∞))by mapping the standard orthonormal basis inl²onto the basis formed by

φn(t) =e^−t/2Ln(t), whereLn denotes the normalizedn-th Laguerre polynomial.

Under this transformationH_λ becomes an integral operator with kernel

K(x,t) = Γ(λ)W_1−λ,1/2(x+t)(x +t)⁻¹, whereW_1−λ,1/2denotes the Whittaker function.

Using some special identities satisfied by these functions Rosenblum proves that this operator has eigenfunctions in L¹((0,∞))of the formx →W_1−λ,µ(x)x⁻¹, which gives latent eigenvectors inl²of the form(x_n)with

x = Z ∞

W (x)φ (x)dx

. (1)

On Rosenblum’s approach

We move froml²toL²((0,∞))by mapping the standard orthonormal basis inl²onto the basis formed by

φn(t) =e^−t/2Ln(t), whereLn denotes the normalizedn-th Laguerre polynomial.

Under this transformationH_λ becomes an integral operator with kernel

K(x,t) = Γ(λ)W_1−λ,1/2(x+t)(x +t)⁻¹, whereW_1−λ,1/2denotes the Whittaker function.

Using some special identities satisfied by these functions Rosenblum proves that this operator has eigenfunctions in L¹((0,∞))of the formx →W_1−λ,µ(x)x⁻¹, which gives latent eigenvectors inl²of the form(x_n)with

x = Z ∞

W (x)φ (x)dx

. (1)

For the spectrum ofH_λonl²he observes that the integral operator constructed above commutes with the symmetric differential operator

f → −(x²f⁰(x))⁰−((1−λ)x −x²/4+1/4)f(x) and proves that this differential operator has a self-adjoint extension whose spectral measure can be found explicitly.

An alternative approach

We identify sequences with power series (convergent in the unit disc)

Under this identificationl²becomes the Hardy spaceH² andH_λ turns into the Hankel operator with symbolz^λ−1if λ /∈N, orz^λ−1logz ifλ∈N.

The latter can also be seen as a limit case of the previous one. To avoid complicated formulas, we will disregard this case for the rest of the talk. The results we obtain cover this case as well.

This operator onH²can be defined on other spaces as well via Cauchy duality.

An alternative approach

We identify sequences with power series (convergent in the unit disc)

Under this identificationl²becomes the Hardy spaceH² andH_λ turns into the Hankel operator with symbolz^λ−1if λ /∈N, orz^λ−1logz ifλ∈N.

The latter can also be seen as a limit case of the previous one. To avoid complicated formulas, we will disregard this case for the rest of the talk. The results we obtain cover this case as well.

This operator onH²can be defined on other spaces as well via Cauchy duality.

An alternative approach

We identify sequences with power series (convergent in the unit disc)

Under this identificationl²becomes the Hardy spaceH² andH_λ turns into the Hankel operator with symbolz^λ−1if λ /∈N, orz^λ−1logz ifλ∈N.

The latter can also be seen as a limit case of the previous one. To avoid complicated formulas, we will disregard this case for the rest of the talk. The results we obtain cover this case as well.

This operator onH²can be defined on other spaces as well via Cauchy duality.

An alternative approach

We identify sequences with power series (convergent in the unit disc)

Under this identificationl²becomes the Hardy spaceH² andH_λ turns into the Hankel operator with symbolz^λ−1if λ /∈N, orz^λ−1logz ifλ∈N.

The latter can also be seen as a limit case of the previous one. To avoid complicated formulas, we will disregard this case for the rest of the talk. The results we obtain cover this case as well.

This operator onH²can be defined on other spaces as well via Cauchy duality.

An alternative approach

We identify sequences with power series (convergent in the unit disc)

Under this identificationl²becomes the Hardy spaceH² andH_λ turns into the Hankel operator with symbolz^λ−1if λ /∈N, orz^λ−1logz ifλ∈N.

The latter can also be seen as a limit case of the previous one. To avoid complicated formulas, we will disregard this case for the rest of the talk. The results we obtain cover this case as well.

This operator onH²can be defined on other spaces as well via Cauchy duality.

H^p ={f holomorphic inD: sup

0<r<1

1 2π

Z 2π

0

f(re^iθ)

p!¹_p

<∞},

A^−τ ={f analytic inD: kfk=sup

z∈D

(1− |z|)^τ|f(z)|<∞},

A^−τ₀ ={f ∈A^−τ : lim

|z|→1(1− |z|)^τ|f(z)|=0}, The intuition is that the latent roots are just eigenvalues of the corresponding operator on spaces larger thanH².

H^p ={f holomorphic inD: sup

0<r<1

1 2π

Z 2π

0

f(re^iθ)

p!¹_p

<∞},

A^−τ ={f analytic inD: kfk=sup

z∈D

(1− |z|)^τ|f(z)|<∞},

A^−τ₀ ={f ∈A^−τ : lim

|z|→1(1− |z|)^τ|f(z)|=0}, The intuition is that the latent roots are just eigenvalues of the corresponding operator on spaces larger thanH².

H^p ={f holomorphic inD: sup

0<r<1

1 2π

Z 2π

0

f(re^iθ)

p!¹_p

<∞},

A^−τ ={f analytic inD: kfk=sup

z∈D

(1− |z|)^τ|f(z)|<∞},

A^−τ₀ ={f ∈A^−τ : lim

|z|→1(1− |z|)^τ|f(z)|=0}, The intuition is that the latent roots are just eigenvalues of the corresponding operator on spaces larger thanH².

H^p ={f holomorphic inD: sup

0<r<1

1 2π

Z 2π

0

f(re^iθ)

p!¹_p

<∞},

A^−τ ={f analytic inD: kfk=sup

z∈D

(1− |z|)^τ|f(z)|<∞},

A^−τ₀ ={f ∈A^−τ : lim

|z|→1(1− |z|)^τ|f(z)|=0}, The intuition is that the latent roots are just eigenvalues of the corresponding operator on spaces larger thanH².

Besides the Hankel form obtained via Cauchy duality,H_λhas well known representations which hold true for every functionf analytic in the disc and integrable on[0,1):

H_λf(z) = Z 1

0

t^λ−1f(t) 1−tz dt,

if Reλ >0 andλ /∈N. For generalλ /∈Nwe can use the formula

H_λf(z) = 1 c

Z

C

f(t)t^λ−1

1−tz dt . (2)

wherec=e^2πiλ−1 andCis a contour inD∪ {1}avoiding the origin, say,C =( the circle of radius ¹₂)+₁

2,1 .

Besides the Hankel form obtained via Cauchy duality,H_λhas well known representations which hold true for every functionf analytic in the disc and integrable on[0,1):

H_λf(z) = Z 1

0

t^λ−1f(t) 1−tz dt,

if Reλ >0 andλ /∈N. For generalλ /∈Nwe can use the formula

H_λf(z) = 1 c

Z

C

f(t)t^λ−1

1−tz dt . (2)

wherec=e^2πiλ−1 andCis a contour inD∪ {1}avoiding the origin, say,C =( the circle of radius ¹₂)+₁

2,1 .

Theorem 1

Letf be analytic inDand integrable on[0,1], and letH_λf be defined by the above integral representation.

Then

i) iff ∈L^p([0,1]), 1<p<∞thenH_λf ∈H^p;

ii) if 0< τ <1 and|f(x)|=O((1−x)^−τ), x →1, then H_λf ∈A^−τ;

iii) if 0< τ <1 and|f(x)|=o((1−x)^−τ), x →1, then H_λf ∈A^−τ₀ .

Theorem 1

Letf be analytic inDand integrable on[0,1], and letH_λf be defined by the above integral representation.

Then

i) iff ∈L^p([0,1]), 1<p<∞thenH_λf ∈H^p;

ii) if 0< τ <1 and|f(x)|=O((1−x)^−τ), x →1, then H_λf ∈A^−τ;

iii) if 0< τ <1 and|f(x)|=o((1−x)^−τ), x →1, then H_λf ∈A^−τ₀ .

Theorem 1

Letf be analytic inDand integrable on[0,1], and letH_λf be defined by the above integral representation.

Then

i) iff ∈L^p([0,1]), 1<p<∞thenH_λf ∈H^p;

ii) if 0< τ <1 and|f(x)|=O((1−x)^−τ), x →1, then H_λf ∈A^−τ;

iii) if 0< τ <1 and|f(x)|=o((1−x)^−τ), x →1, then H_λf ∈A^−τ₀ .

Theorem 1

Letf be analytic inDand integrable on[0,1], and letH_λf be defined by the above integral representation.

Then

i) iff ∈L^p([0,1]), 1<p<∞thenH_λf ∈H^p;

ii) if 0< τ <1 and|f(x)|=O((1−x)^−τ), x →1, then H_λf ∈A^−τ;

iii) if 0< τ <1 and|f(x)|=o((1−x)^−τ), x →1, then H_λf ∈A^−τ₀ .

Corollary

IfX is one of the spacesH^p,1<p<∞,A^−τ orA^−τ₀ ,0< τ <1, thenH_λis a bounded linear operator fromX into itself.

Moreover,H_λ is the unique operatorT onX with Tzⁿ=

∞

X

m=0

z^m m+n+λ,

where by abuse of notation we writezfor the identity function onD.

Corollary

IfX is one of the spacesH^p,1<p<∞,A^−τ orA^−τ₀ ,0< τ <1, thenH_λis a bounded linear operator fromX into itself.

Moreover,H_λ is the unique operatorT onX with Tzⁿ=

∞

X

m=0

z^m m+n+λ,

where by abuse of notation we writezfor the identity function onD.

Corollary

IfX is one of the spacesH^p,1<p<∞,A^−τ orA^−τ₀ ,0< τ <1, thenH_λis a bounded linear operator fromX into itself.

Moreover,H_λ is the unique operatorT onX with Tzⁿ=

∞

X

m=0

z^m m+n+λ,

where by abuse of notation we writezfor the identity function onD.

Differential operators in the commutator

We want to find differential operators which formally commute withH_λ.

Df =q₃f⁰⁰+q₂f⁰+q₁f (3) where

q₃(z) =

3

X

0

α_jz^j , q₂(z) =

2

X

0

β_jz^j , q₁(z) =γ₁z .

There is a family of such operators for whichDH_λ−H_λDhas rank one.

Theorem 2

LetDbe a differential operator of the form (3). Assume that the polynomialsq₁,q₂,q₃satisfy:

q₁(z) = [αλ(λ+1)−λγ]z

q₂(z) = (z−1)[(2α(λ+1)−γ)z−γ] q₃(z) =αz(z−1)²

for some constantsα, γ ∈C. Then for everyf ∈A^−τ, (0< τ <1), we have

H_λDf −DH_λf = 1

c(λ−1)(αλ−γ) Z

C

f(t)t^λ−2dt

whereCis the contour in (2).

Corollary Let

D_1,λf(z) = (z²−1)f⁰(z) +λzf(z),

D_2,λf(z) =z(z−1)²f⁰⁰(z) + (z−1)[(λ+2)z−λ]f⁰(z) +λzf(z). Then

H_λD_1,λ−D_1,λH_λ = ¹_c(λ−1)R

Cf(t)t^λ−2dt , H_λD_2,λ=D_2,λH_λ .

Corollary Let

D_1,λf(z) = (z²−1)f⁰(z) +λzf(z),

D_2,λf(z) =z(z−1)²f⁰⁰(z) + (z−1)[(λ+2)z−λ]f⁰(z) +λzf(z). Then

H_λD_1,λ−D_1,λH_λ = ¹_c(λ−1)R

Cf(t)t^λ−2dt , H_λD_2,λ=D_2,λH_λ .

Corollary Let

D_1,λf(z) = (z²−1)f⁰(z) +λzf(z),

D_2,λf(z) =z(z−1)²f⁰⁰(z) + (z−1)[(λ+2)z−λ]f⁰(z) +λzf(z). Then

H_λD_1,λ−D_1,λH_λ = ¹_c(λ−1)R

Cf(t)t^λ−2dt , H_λD_2,λ=D_2,λH_λ .

Corollary Let

D_1,λf(z) = (z²−1)f⁰(z) +λzf(z),

D_2,λf(z) =z(z−1)²f⁰⁰(z) + (z−1)[(λ+2)z−λ]f⁰(z) +λzf(z). Then

H_λD_1,λ−D_1,λH_λ = ¹_c(λ−1)R

Cf(t)t^λ−2dt , H_λD_2,λ=D_2,λH_λ .

Proposition

1 Forν ∈Cthe space of solutions of the equation D_1,λf =νf

is one-dimensional and it is spanned by f_λ(z) = (z−1)^−λ+ν2 (z+1)^−λ−ν2 .

2 The solutions of the equation

D_1,λg−νg=f_λ

are given by g_λ(z) =

1 2log

z−1 z+1

+k

(z−1)^−λ+ν2 (z+1)^−λ−ν2 ,

Proposition

1 Forν ∈Cthe space of solutions of the equation D_1,λf =νf

is one-dimensional and it is spanned by f_λ(z) = (z−1)^−λ+ν2 (z+1)^−λ−ν2 .

2 The solutions of the equation

D_1,λg−νg=f_λ

are given by g_λ(z) =

1 2log

z−1 z+1

+k

(z−1)^−λ+ν2 (z+1)^−λ−ν2 ,

Proposition

1 Forν ∈Cthe space of solutions of the equation D_1,λf =νf

is one-dimensional and it is spanned by f_λ(z) = (z−1)^−λ+ν2 (z+1)^−λ−ν2 .

2 The solutions of the equation

D_1,λg−νg=f_λ

are given by g_λ(z) =

1 2log

z−1 z+1

+k

(z−1)^−λ+ν2 (z+1)^−λ−ν2 ,

Remark

Due to the integral representation H_λf(z) = 1

c Z

C

f(t)t^λ−1

1−tz dt. (4)

the eigenfunctions ofH_λare analytic near−1.

IfH_λhas a finite dimensional eigenspace which is invariant forD_1,λthen it is spanned by functions of the form

fn(z) = (z−1)^−n−λ(z+1)ⁿ, n≥0.

Remark

Due to the integral representation H_λf(z) = 1

c Z

C

f(t)t^λ−1

1−tz dt. (4)

the eigenfunctions ofH_λare analytic near−1.

IfH_λhas a finite dimensional eigenspace which is invariant forD_1,λthen it is spanned by functions of the form

fn(z) = (z−1)^−n−λ(z+1)ⁿ, n≥0.

Remark

Due to the integral representation H_λf(z) = 1

c Z

C

f(t)t^λ−1

1−tz dt. (4)

the eigenfunctions ofH_λare analytic near−1.

IfH_λhas a finite dimensional eigenspace which is invariant forD_1,λthen it is spanned by functions of the form

fn(z) = (z−1)^−n−λ(z+1)ⁿ, n≥0.

The eigenvalue problem forD_2,λis considerably harder sinceD_2,λ has order two andD_2,λdoes not preserve the degree of polynomials.

The equation

D_2,λf =νf

is an equation of Heun’s type and it can be reduced to the hypergeometric equation (many thanks to Boris Shapiro)

z(1−z)f⁰⁰+ [γ−(α+β+1)z]f⁰−αβf =0.

The eigenvalue problem forD_2,λis considerably harder sinceD_2,λ has order two andD_2,λdoes not preserve the degree of polynomials.

The equation

D_2,λf =νf

is an equation of Heun’s type and it can be reduced to the hypergeometric equation (many thanks to Boris Shapiro)

z(1−z)f⁰⁰+ [γ−(α+β+1)z]f⁰−αβf =0.

As usual we denote the hypergeometric function by

2F₁(α, β;γ;z) =1+ αβ

1!γz+ α(α+1)β(β+1)

2!γ(γ+1) z²+...

=

∞

X

n=0

(α)n(β)n

(γ)_n zⁿ n!,

where(α)_n=α(α+1). . .(α+n−1).

Theorem 3

Forλ∈C\ {0,−1, . . .}andν ∈C, the solutions of the eigenvalue problem

D_2,λf =νf

analytic inDform a one-dimensional space spanned by f(z) = (z−1)^a₂F₁(a+1,a+λ;λ;z)

= (z−1)^a0 ₂F₁(a⁰+1,a⁰+λ;λ;z), whereaanda⁰ are solutions of the quadratic equation

x²+x+λ=ν .

Two nontrivial questions arise:

1 When are the hypergeometric functions given in Theorem 3 in the spaces considered here?

2 If so, what is the corresponding eigenvalue ofH_λ?

Two nontrivial questions arise:

1 When are the hypergeometric functions given in Theorem 3 in the spaces considered here?

2 If so, what is the corresponding eigenvalue ofH_λ?

Theorem 4

Letλ∈C\ {0,−1, , . . .}. Fora∈Clet

f_a(z) = (z−1)^a₂F₁(a+1,a+λ;λ,z). Then, for−¹₂ ≥ <a>−1,H_λfais defined and satisfies

H_λfa=− π sinπafa.

Moreover,

i) Ifa6=−¹₂anda+1−λ /∈N∪ {0}thenf_a∈A^−τ iffτ >−<a andfa∈H^piff ¹_p ><a.

ii) Ifa6=−¹₂andλ−a−1=−n,n∈N∪ {0}thenf_a∈A^−τ for allτ ≥ <a+1,fa∈A^−τ₀ for allτ ><a+1 andfa∈H^p whenever ¹_p ><a+1.

iii) Ifa=−¹₂ thenfa∈A^−τ₀ forτ > ¹₂ andfa∈H^pwhenever p<2.

Moreover,

i) Ifa6=−¹₂anda+1−λ /∈N∪ {0}thenf_a∈A^−τ iffτ >−<a andfa∈H^piff ¹_p ><a.

ii) Ifa6=−¹₂andλ−a−1=−n,n∈N∪ {0}thenf_a∈A^−τ for allτ ≥ <a+1,fa∈A^−τ₀ for allτ ><a+1 andfa∈H^p whenever ¹_p ><a+1.

iii) Ifa=−¹₂ thenfa∈A^−τ₀ forτ > ¹₂ andfa∈H^pwhenever p<2.

Moreover,

i) Ifa6=−¹₂anda+1−λ /∈N∪ {0}thenf_a∈A^−τ iffτ >−<a andfa∈H^piff ¹_p ><a.

ii) Ifa6=−¹₂andλ−a−1=−n,n∈N∪ {0}thenf_a∈A^−τ for allτ ≥ <a+1,fa∈A^−τ₀ for allτ ><a+1 andfa∈H^p whenever ¹_p ><a+1.

iii) Ifa=−¹₂ thenfa∈A^−τ₀ forτ > ¹₂ andfa∈H^pwhenever p<2.

Moreover,

i) Ifa6=−¹₂anda+1−λ /∈N∪ {0}thenf_a∈A^−τ iffτ >−<a andfa∈H^piff ¹_p ><a.

ii) Ifa6=−¹₂andλ−a−1=−n,n∈N∪ {0}thenf_a∈A^−τ for allτ ≥ <a+1,fa∈A^−τ₀ for allτ ><a+1 andfa∈H^p whenever ¹_p ><a+1.

iii) Ifa=−¹₂ thenfa∈A^−τ₀ forτ > ¹₂ andfa∈H^pwhenever p<2.

Rosenblum found one or two eigenvalues ofH_λ onH² which may have multiplicity greater than one.

For spaces larger thanH²an open set of eigenvalues with multiplicity one appears.

The corresponding eigenfunctions are hypergeometric functions (generalized Legendre functions of the first kind).

Rosenblum found one or two eigenvalues ofH_λ onH² which may have multiplicity greater than one.

For spaces larger thanH²an open set of eigenvalues with multiplicity one appears.

The corresponding eigenfunctions are hypergeometric functions (generalized Legendre functions of the first kind).

Rosenblum found one or two eigenvalues ofH_λ onH² which may have multiplicity greater than one.

For spaces larger thanH²an open set of eigenvalues with multiplicity one appears.

The corresponding eigenfunctions are hypergeometric functions (generalized Legendre functions of the first kind).

Which are all the eigenvalues ofH_λ, on the spaces considered?

In particular, how can one find Rosenblum’s eigenvalues?

Interesting to note that those arise from the analysis of the commutation relation

H_λD_1,λ−D_1,λH_λ = 1

c(λ−1) Z

C

f(t)t^λ−2dt .

Which are all the eigenvalues ofH_λ, on the spaces considered?

In particular, how can one find Rosenblum’s eigenvalues?

Interesting to note that those arise from the analysis of the commutation relation

H_λD_1,λ−D_1,λH_λ = 1

c(λ−1) Z

C

f(t)t^λ−2dt .

Which are all the eigenvalues ofH_λ, on the spaces considered?

In particular, how can one find Rosenblum’s eigenvalues?

Interesting to note that those arise from the analysis of the commutation relation

H_λD_1,λ−D_1,λH_λ = 1

c(λ−1) Z

C

f(t)t^λ−2dt .

Which are all the eigenvalues ofH_λ, on the spaces considered?

In particular, how can one find Rosenblum’s eigenvalues?

Interesting to note that those arise from the analysis of the commutation relation

H_λD_1,λ−D_1,λH_λ = 1

c(λ−1) Z

C

f(t)t^λ−2dt .

LetX be one of the spacesH^p,p>1, orA^−τ,0< τ <1.

Main Result

Letλ∈C\ {0,−1, , . . .}and let

S(X) ={a: (z−1)^a∈X}.

Main Result

i)Ifλ∈ S(X),H_λ has the eigenvalues±πcscπλwith multiplicities_N

2

and_N−1

2

respectively, whereNis the largest integer for which the functionz →(z−1)^−N−λ belongs toX. Moreover, if

f_n(z) = (z−1)^−n−λ(z+1)ⁿ, 0≤n≤N, then ker(H_λ−πcscπλI)is spanned by the functionsf_2k, 0≤k ≤^N₂

and ker(H_λ+πcscπλI)is spanned by the functionsf_2k+1, 0≤k ≤^N−1₂ .

Main Result

ii)Ifp≥2 or 0< τ < ¹₂,H_λ has no other eigenvalues.

Main Result

iii)Ifp<2 or ¹₂ < τ <1, the point spectrum ofH_λ onX is the image of the set

S(X)\

<a≤ −1 2

∪ {−λ,−λ−1}

by the mapa→ −πcscπa. Each eigenvalue−πcscπa, a6=−λ,−λ−1, has multiplicity one and the corresponding eigenspace is spanned by the hypergeometric function (z−1)^a₂F₁(a+1,a+λ;λ;·).

Finally, ifX =A^−1/2, the point spectrum ofH_λ contains the image of the set

S(A^−1/2)\

<a<−1 2

∪ {−λ,−λ−1}

\

−1 2

A few words about the proof The argument detects two types of eigenspaces:

1 Eigenspaces where the restrictions ofH_λandD_1,λ

commute, these are those found by Rosenblum in the case whenX =H²andλ∈R.

2 Eigenspaces where the restrictions ofH_λandD_1,λdo not commute.

These appear only in spaces strictly larger thanH²and the corresponding eigenvalues form an open set.

A few words about the proof The argument detects two types of eigenspaces:

1 Eigenspaces where the restrictions ofH_λandD_1,λ

commute, these are those found by Rosenblum in the case whenX =H²andλ∈R.

2 Eigenspaces where the restrictions ofH_λandD_1,λdo not commute.

These appear only in spaces strictly larger thanH²and the corresponding eigenvalues form an open set.

A few words about the proof The argument detects two types of eigenspaces:

1 Eigenspaces where the restrictions ofH_λandD_1,λ

commute, these are those found by Rosenblum in the case whenX =H²andλ∈R.

2 Eigenspaces where the restrictions ofH_λandD_1,λdo not commute.

These appear only in spaces strictly larger thanH²and the corresponding eigenvalues form an open set.

All eigenspaces have finite dimension!

Outline of the proof

IfH_λ has an infinite dimensional eigenspace thenH_λ−2has an infinite dimensional eigenspace which contains a sequencefn

such that

kf_nk₂=1, sup_n

(1−z)⁻¹fn

₂<∞, fnhas a zero of ordernat the origin.

All eigenspaces have finite dimension!

Outline of the proof

IfH_λ has an infinite dimensional eigenspace thenH_λ−2has an infinite dimensional eigenspace which contains a sequencefn

such that

kf_nk₂=1, sup_n

(1−z)⁻¹fn

₂<∞, fnhas a zero of ordernat the origin.

All eigenspaces have finite dimension!

Outline of the proof

IfH_λ has an infinite dimensional eigenspace thenH_λ−2has an infinite dimensional eigenspace which contains a sequencefn

such that

kf_nk₂=1, sup_n

(1−z)⁻¹fn

₂<∞, fnhas a zero of ordernat the origin.

All eigenspaces have finite dimension!

Outline of the proof

IfH_λ has an infinite dimensional eigenspace thenH_λ−2has an infinite dimensional eigenspace which contains a sequencefn

such that

kf_nk₂=1, sup_n

(1−z)⁻¹fn

₂<∞, fnhas a zero of ordernat the origin.

Then by the Hankel-type identity 1

zⁿH_λ−2f_n(z) = (H_λ−2ζⁿf_n) (z) we obtain

µf_n(z)

zⁿ = (H_λ−2ζⁿf_n) (z).

Then by the Hankel-type identity 1

zⁿH_λ−2f_n(z) = (H_λ−2ζⁿf_n) (z) we obtain

µf_n(z)

zⁿ = (H_λ−2ζⁿf_n) (z).

By the integral representation and duality

|µ|=

µf_n(z) zⁿ

2

≤ sup

khk₂≤1

1 c

Z

C

fn(t)h(t)t^λ+n−3dt

= sup

khk₂≤1

1 c

Z

C

(1−t)fn(t)

1−th(t)t^λ+n−3dt

≤ 1

|c|

Z

C

t^λ+n−3

|dt| →0 whenn→ ∞ the last inequality is due to

f_n(t)h(t) 1−t

≤ k 1−t

f_n 1−t

₂

khk₂.